# Informal proof of Godel's second incompleteness theorem

This relates to two previous threads:

Question about Godel's first incompleteness theorem and the theory within which it is proved

Explanation of proof of Gödel's Second Incompleteness Theorem (I'm using ideas from the proof given here.)

I'd like to know if the following informal proof of Godel's 2nd incompleteness is correct. We accept Godel's 1st incompleteness theorem as proven:

1. We have a theory $\sf{T}$ capable of basic arithmetic.

2. Theory $\sf{T}$ is capable of proving Godel's 1st incompleteness theorem. (I'm suspicious about this)

3. From 2, Theory $\sf{T}$ is capable of proving the following statement about itself: "If $\sf{T}$ is consistent, then $\sf{T}$'s godel statement $G$ is true but unprovable within $\sf{T}$"

4. Assume T proves its own consistency.

5. Then using 3 and 4, it proves its own godel statement $G$ is true but unprovable within T.

6. Since $\sf{T}$ proves the godel statement $G$ is true... $\sf{T}$ can assert "$G$ is provable within $\sf{T}$" (suspicious?)...

7. Using 5 and 6, $\sf{T}$ asserts $G$ is both provable and unprovable within $\sf{T}$, so $\sf{T}$ is inconsistent.

Is this essentially correct? Thanks in advance.